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I'm writing a mathematical paper. In it, I use a lemma. The lemma is not hard to prove and I have verified it myself. The proof is too tedious to include in the paper, so I want to just include a citation. I found a paper that includes the result. However, that paper does not actually include a proof. I cannot find any other place where this lemma appears.

I see three options:

  1. State the lemma without proof or citation.

  2. State the lemma without proof, but cite the paper (that states the lemma without proof or citation).

  3. Provide a proof of the lemma.

Which is most appropriate? Option 1 is easiest, but might annoy some readers who don't believe me. Option 2 seems like a cop out. Option 3 is safest, but I don't think it's necessary, as the proof is really just a long and boring calculation.

ADDED: To be clear, the lemma is basically an integral. The proof consists of splitting up the domain of integration to remove absolute values, evaluating each of the parts (easy enough for symbolic integration packages like mathematica), and then joining them back up. This is "obvious", but messy because the expressions are quite long. My writeup is two pages.

Maybe a better way to phrase my question: The result is trivial -- I think so, the authors of the other paper think so, and the journal they published in thinks so. Should I still provide a citation? Is it misleading to cite the other paper without clarifying that it doesn't provide a proof?

user80085
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  • It sounds like you used the lemma first and then searched for other appearances of it in publications. Is this the case? If so, did you create it yourself or did you obtain it from somewhere? – Mick Sep 20 '17 at 04:00
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    "If only I had the theorems! Then I should find the proofs easily enough." -Riemann . If someone states a lemma/theorem/conjecture you use (even if you reinvented it), you need to cite the source regardless if there is a proof or not. If you easily found a proof, then you should probably be convinced (as long as the original paper acted like it was a fact) that the original writer also knew a proof. Saying that if you use something and the proof isn't widely available, you are probably doing a disservice to the reader to not include it(unless you can show the reader how to reproduce a proof). – PVAL Sep 20 '17 at 04:05
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    If not for providing evidence, you should at least provide a proof for making the life of your readers easier. However, you have multiple options if you don't want to include it in the main part of your paper. – koalo Sep 20 '17 at 06:36
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    Chemical papers often make use of Supporting Information for these very things: parts the main paper builds on but too irrelevant to be included in it. – Jan Sep 20 '17 at 13:18

6 Answers6

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Citing the other paper seems necessary in any case, as they have stated the lemma before you. This is for attribution. Citing the other paper for evidence seems not appropriate, as there is no proof given there.

If the lemma is not rather obvious (say the obvious proof strategy works in < 5min), then stating it without proof would be very bad form. Put in an appendix if you don't want a boring lengthy proof to spoil the otherwise elegant paper, but put it somewhere people can find it.

Arno
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    The obvious retort is "If they can state it without proof, then why can't I?" – user80085 Sep 19 '17 at 22:37
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    Well, if they had written down the proof as they should have, you wouldnt have this problem now, right? – Arno Sep 19 '17 at 22:54
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    I feel like citing a peer-reviewed paper should be sufficient evidence, no? If the reader is really that skeptical, they will be able to re-derive the result. – user80085 Sep 19 '17 at 22:54
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    Peer review means very little - and if there is no proof for that lemma in the paper, the reviewers cant have checked the proof (of course they could have derived a proof themselves, but how many more peole are supposed to do that?). – Arno Sep 19 '17 at 23:00
  • I guess what I'm saying is that, they (and their reviewers) thought it was trivial enough to omit the proof, and so do I. So I'm inclined to omit the proof. However, it feels odd to cite a source for a result that doesn't actually provide a proof. – user80085 Sep 19 '17 at 23:09
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    Agree with Arno. I don't see any disadvantage in including the proof. This is the only professional solution here: prove it and cite the lemma. I would also add a note "X stated the lemma without a proof, and thus we shall prove it here", which will sound slightly critical of the original authors, as it should be. It seems user80085 believes too much in the system, no offense, as if the fact that a paper passed peer-review means every aspect of the paper is justified. It's obviously not. – Dilworth Sep 19 '17 at 23:30
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    Given that two people have now used this lemma, I think a proof is worth presenting even if it is "trivial". You can send in your writeup (which you say you already have) as an appendix with this paper, and if the reviewers also feel it doesn't belong in this paper, post it as a short paper in arXiv/etc. so the third user of this lemma won't face your problem. – Jeffrey Bosboom Sep 20 '17 at 00:55
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    It's not clear that attribution means a whole lot in this case, which I suspect is the whole reason that the question is being asked in the first place. I think it would be helpful to justify the need for attribution, rather than just asserting it. After all, by the classical definition of knowledge as justified true belief, the original authors didn't "know" the result (it's true but they didn't demonstrate any justification), so why do they deserve to have it attributed to them? Isn't an attribution to a result stated without proof just an appeal to authority? – David Richerby Sep 20 '17 at 15:05
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    @JeffreyBosboom It probably won't do as a paper on the arXiv if it's just a lemma. However, it could be attached to the main paper on the arXiv. – Jessica B Sep 21 '17 at 14:21
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    @user80085: Look at it this way: if you include the proof of the lemma, the next person to use it will likely cite your proof. More citations for very little work on your part == win-win scenario. – tonysdg Sep 21 '17 at 15:21
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    @Dilworth: I have seen referee reports where the referee was rather annoyed that a long proof of a rather trivial statement was included. Also, the length of a paper is sometimes a deciding factor regarding possible journals, as there are many that have a required minimum of maximum number of pages. Annoying a referee and not fitting within a journal's specifications are definitely "disadvantages". – Martin Argerami Sep 22 '17 at 14:49
  • @MartinArgerami: your point is indeed valid. One can write a proof in the arXiv preprint, omit it in the submitted version and explain in the cover letter why the proof was omitted. Then the proof is available, and the page-length constraints we still drag with us will not play against the paper. – Benoît Kloeckner Sep 22 '17 at 21:05
  • @MartinArgerami, this is an anecdotal evidence only. The risk to annoy a referee due to a "redundant" proof is much smaller than the risk of getting rejected or annoy a referee due to omission of necessary proofs. In the former case, the referee can simply ask to take the proof out. – Dilworth Sep 23 '17 at 22:20
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    @Dilworth: of course it is anecdotal; do you know of any scientific study of referees attitudes with respect to long proofs of trivial lemmas? If the referee "asks to take them out", that usually belongs to a "major revision". And if you think that a "major revision" looks the same on an editor's eye as a "recommend publication", you are being naive. So, as I said, it is a disadvantage. – Martin Argerami Sep 23 '17 at 22:20
  • @MartinArgerami, no. The opposite of "anecdotal" here is not a systematic research, rather an experience of many incidents in which papers are rejected due to "unnecessary proofs". Also, taking out redundant computations does not imply a major revision. Further, as I said there's a risk also in omitting proofs, so for your argument to hold, you have to argue that the risk in omission is lower than in inclusion of proofs. – Dilworth Sep 23 '17 at 22:23
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If the result is basically trivial (as you say it is), I think how you proceed should consider how standard this type of result would be in the field.

You could put something like:

The following result can be established by standard (but tedious) computation.

if it's the sort of thing you could expect an early graduate student in the field to do as a homework question, or

The following result, which is stated by (Author) in [(paper)], can be established by standard (but tedious) computation.

otherwise.

Jessica B
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    The amount of discussion and contemplation on whether to prove "trivial lemmas" already shows why any professional mathematician should include the proof of these "trivialities", and redeem us from such discussions. Indeed, if it's so trivial, why not just prove it and finish with it? – Dilworth Sep 21 '17 at 22:51
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    @Dilworth I don't really agree. I don't think citing something trivial is actually a good thing. If it's trivial, the person who wrote it down doesn't really own it in the same way as a non-trivial result. Also, it's not obvious where you would look for a trivial result, as it could appear in papers on a range of topics. Moreover, often if the result is trivial there is not just one version of it. I think whether you include a proof of the exact variant needed for your paper will depend on how crucial the tiny details are to you. – Jessica B Sep 22 '17 at 05:39
  • @Dilworth On the other hand, a useful thing a mathematician could do in such circumstances is write a textbook that includes a proof. But of course the incentive to do so it lower. Alternatively, I like the idea of papers in electronic form where most of the detail is initially hidden and you can expand out more layers as you wish. Then the proof could be included for those that want to read it, without getting in the way for those who don't. – Jessica B Sep 22 '17 at 05:42
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    Don't say "by standard techniques" say which standard techniques you're using! – Noah Snyder Sep 22 '17 at 13:19
  • @NoahSnyder The OP has said this is an integral. You don't need to say that you calculate an integral using standard integration techniques. – Jessica B Sep 22 '17 at 18:35
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    @JessicaB, I was not talking about citation indeed. I was talking about proving what you actually state is true. Papers are written not just to inform us of new results, but to validate and establish true statements. Working out tedious details to yield important results is a contribution by itself. – Dilworth Sep 23 '17 at 22:40
  • @Dilworth You said 'prove it and finish with it'. So you expect others not to have to prove it again, because they can in future cite the version you have written. I am saying I do not think that is a good approach. – Jessica B Sep 24 '17 at 07:55
  • @JessicaB, I'm saying that you must prove or refer to a proof of any statement you state is true. I didn't say you are not supposed to cite the original (unproved) statement. If people in future cite you as the one proving the claim rather the original unproved statement, this is a bonus (teaching a lesson to the original authors). – Dilworth Sep 24 '17 at 13:09
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Cite the paper when you state the lemma. Then write:

\begin{proof} Split up the domain of integration to remove absolute values, then evaluate each of the parts. \end{proof}

It's a waste of everyone's time to have two pages of a calculus exercise. But it's also a waste of everyone's time to have to guess how the proof goes. The above is the best compromise that makes it clear how the proof goes in the least amount of time.

If the proof were one paragraph instead of two pages then I'd say include it all.

Noah Snyder
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  • This was my thought also, although perhaps with slightly more explanation if some tricky manipulations are involved (but the OP's description suggests this is not the case), and perhaps also state the result of the evaluation for each of the parts. In addition, I think it would be useful cite the other reference at the beginning and say something to the effect that you (the OP) have confirmed the result stated in [XX] by using the following method, and then give the brief explanation. – Dave L Renfro Sep 22 '17 at 17:31
  • Yeah, I don't know the actual calculation. Perhaps another sentence or two would be in order like: "for the third part we use integration by parts with u=blah and dv=blah." But at any rate it seems clear that two pages is too much, but zero sentences is too little. – Noah Snyder Sep 22 '17 at 20:19
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Just say you have discovered a truly marvelous proof for it that won't fit in the confines of your paper's length restrictions. ;)

No one will mind, right? They can always work out the proof that you had in mind....

I recommend including a proof in the appendix, if none has previously been published.

A proof without proof is just a statement. If you feel you shouldn't just state something without any proof at all, then don't state "there is a proof" without any proof of that statement.

The historical example I've alluded to is a good illustration of the problems that can arise from the unproven assertion, "I have a proof for this."

Wildcard
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    This is just a joke, followed by a repetition of a suggestion that had already been posted when this answer was written. – David Richerby Sep 21 '17 at 10:10
  • @DavidRicherby the joke has a point, though. If you think about the parallel I've drawn, you will gain your own insight into the perils of asserting the existence of a proof without...well, without proof. (Edit: I've made this more explicit in my answer.) – Wildcard Sep 21 '17 at 10:34
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I suggest not including the proof in your paper.

If you need to, cite the other paper which states the lemma Then, since you have determined that the proof is "obvious", simply state that.

For example;

  • "Lemma 2 is stated without proof by Bloggs (2007). The proof is trivial and not included here" [The wording "and not included here" is optional, since you won't provide a proof];
  • (If you want to provide some pointer on how to start the proof) "Lemma 2 is stated without proof by Bloggs (2007). If one starts by splitting up the domain of integration to remove absolute values, the proof is trivial."

Any competent mathematician will understand your point, since it is fairly common practice in mathematical journals.

If they so desire, the reader will be able to derive the lemma on their own. In fact, some mathematicians will enjoy doing exactly that as an exercise - why deprive them of that enjoyment?

Peter
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    A mathematical paper is normally written for a target audience of mathematicians - or, at least, people with an interest and appreciation of mathematics. It is not necessary to assume a completely uninformed reader, nor is it necessary to spoon-feed the reader. – Peter Sep 22 '17 at 09:07
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    I was talking about mathematicians as well. Many mathematicians are "incompetent" in many aspects of mathematics, as they are only human. Writing papers assuming a priori that everyone obviously knows the basics of "Semi-periodic-C*-elimination-theory" is simply bad writing. – Dilworth Sep 23 '17 at 22:32
  • The OP described a pretty simple proof based on elementary calculus, not a proof relying on obscure theory that will only be understood by a small number of mathematicians. – Peter Sep 23 '17 at 22:40
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    to be fair, the OP added this explanation on the actual proof he refers to, only after the discussion here took place. – Dilworth Sep 23 '17 at 22:42
  • If you want to be fair, my answer was written at least two days after the OP added the explanation. – Peter Sep 23 '17 at 23:13
  • Perhaps. My point still holds. Even routine calculations should be provided in journal versions. First, because what is 'routine' for one person is many times non trivial to another, and secondly, the article should be verifiable easily. If routine calculations are so easy, they could easily be included as well. If they are long and tedious then they are not easy to verify, and so omitting them means you submit a paper without a full proof, only vouching for its correctness, based on the belief that you have done the actual tedious work behind the scene. – Dilworth Sep 24 '17 at 00:13
  • I disagree with your point. The point of a mathematical paper is generally to impart some substantive (e.g. new) knowledge. Adding additional pages of trivial proofs (which, among other things, are easily verifiable) or minor calculations does not increase substantive content. It increases thud factor (e.g. the noise made when the printed paper is dropped on a desk) and the time needed for readers to wade through the paper to find the substantive content. That makes the paper less useful overall. There is value in brevity, and leaving out extraneous information. – Peter Sep 24 '17 at 00:39
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    I disagree. There is value/benefit in brevity just like there is a value/benefits in full proofs. The question is which is more valuable, and what is more appropriate to a journal publication. A journal publication must be correct, leaving out long calculations means there is good chance the authors made a mistake, the reviewer didn't bother to check, and future readers will be unable to follow the proofs. The possible value in brevity is thus smaller than the cons. I also claim that "substantive knowledge" is a relative term, and cannot solely be the point of a mathematical paper. – Dilworth Sep 24 '17 at 13:02
  • You have a completely backward view of how errors get introduced into published materials. Unnecessary inclusion of long calculations or trivial proofs increases the likelihood of error being made (e.g. more chance of typos by the author, by the typesetter, etc) and then remain undetected for longer (author when proof-reading, the reviewer, and eventual reader all are more likely to skim over tedious stuff, or [if they are familiar with the subject] read what they expect to see). The number of errors, and the time errors remain undetected therefore both increase with size of the article. – Peter Sep 24 '17 at 13:12
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    I disagree, factually. My substantial knowledge of errors that made their way into peer-review publication have always been the result of not including the proofs of "obvious/trivial" lemmas. Not the other way around, as you suggest. Making the effort to prove everything reduces substantially the chance of errors. – Dilworth Sep 24 '17 at 13:16
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Could you just give the reader an idea of the tediousness by showing just a part of the proof? Maybe showing the verbosity of just one part of it is the best way to convince any reader that your omission is fully understandable and justified.

Alternatively, if feasible, you could link to an URL containing a (sketch of?) proof, either in LaTeX or, e.g., as a Mathematica nb file.

I would cite the other paper in any case, for attribution, as suggested by others.

anon
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